Number Play · Lesson 7 of 10
Rhythms and the Virahāṅka–Fibonacci Numbers
“Count rhythms of short and long syllables without missing an arrangement.”
• Represent short and long syllables as one-beat and two-beat blocks. • List ordered ways of making a small beat total using 1s and 2s. • Split all rhythms into those beginning with 1 and those beginning with 2. • Explain why consecutive rhythm counts combine to make the next count. • Connect the sequence to its origins in studies of poetry.
Poetry becomes a counting puzzle
In many Indian poetic traditions a short syllable takes one beat and a long syllable takes two beats. A five-beat rhythm might be short–short–long–short, represented by 1 + 1 + 2 + 1. A different order gives a different sound: 1 + 2 + 1 + 1 and 2 + 1 + 1 + 1 are separate rhythms even though each totals five.
An ordered arrangement of 1-beat and 2-beat syllables whose beat lengths total n.
Problem
How many rhythms can be made using exactly three beats?
- 1.Start with three short syllables: 1 + 1 + 1.
- 2.Use one long and one short syllable in both possible orders: 1 + 2 and 2 + 1.
- 3.There are three rhythms. We count orders separately.
A method that cannot miss a rhythm
There is one one-beat rhythm, [1]. There are two two-beat rhythms, [1,1] and [2]. The counts for one through four beats are 1, 2, 3, 5. To find every five-beat rhythm, separate the possibilities by their first syllable. If it is 1, the remaining four beats can follow any four-beat rhythm. If it is 2, the remaining three beats can follow any three-beat rhythm. These two groups do not overlap and include every possibility.
| Beat total | Count | Reason |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 1+1 or 2 |
| 3 | 3 | 2+1 |
| 4 | 5 | 3+2 |
| 5 | 8 | 5+3 |
| 6 | 13 | 8+5 |
| 8 | 34 | 21+13 |
Problem
How many five-beat rhythms are there without listing all eight individually?
- 1.There are five ways to make the remaining four beats after an opening 1.
- 2.There are three ways to make the remaining three beats after an opening 2.
- 3.The first syllable cannot be anything else, so the two nonoverlapping groups give 5 + 3 = 8.
Problem
How many six-beat rhythms are there?
- 1.An opening 1 leaves five beats, which can be made in 8 ways.
- 2.An opening 2 leaves four beats, which can be made in 5 ways.
- 3.Together they give 8 + 5 = 13 six-beat rhythms.
This counting rule produces 1, 2, 3, 5, 8, 13, 21, 34, … . The chapter calls these the Virahāṅka–Fibonacci numbers. It recounts how Virahāṅka described the poetic counting method, notes earlier work of Piṅgala and later discussions by Gopala and Hemachandra, and explains the later name associated with Fibonacci. The mathematics here is the method: count every arrangement by its first part.
The sums 1 + 2 and 2 + 1 have the same numerical total, but they represent different rhythms. Do not merge them when counting.
Quiz
How many beats does a long syllable occupy in this model?
Which two expressions represent different three-beat rhythms?
How many four-beat rhythms are there?
Why is the five-beat count 8?
How many six-beat rhythms are there?
Practice Problems
- Write all five different four-beat rhythms using 1s and 2s.
- List all eight five-beat rhythms. Group them by whether they begin with 1 or 2.
- Explain in words why the two first-syllable groups cannot contain the same rhythm.
- Use the method to find the seven-beat count from the five- and six-beat counts.
- Find how many ways an eight-step staircase can be climbed with steps of size 1 or 2. Explain the connection.
- A classmate counts 1 + 2 and 2 + 1 as one rhythm. Explain the mistake with a spoken beat pattern.
Key Takeaways
• One beat represents a short syllable; two beats represent a long one. • Rhythm order matters even when two expressions have the same total. • Every rhythm starts with either a 1 or a 2, leaving a smaller rhythm to count. • The count for a beat total is the sum of the counts for the two preceding totals. • The Virahāṅka–Fibonacci sequence begins 1, 2, 3, 5, 8, 13, 21, 34.